Skip to main content

Cesàro summable difference sequence space

An Erratum to this article was published on 09 January 2014

Abstract

The difference sequence spaces c 0 (Δ), c(Δ) and (Δ) were introduced by Kizmaz (Can. Math. Bull. 24:169-176, 1981). In this paper, we introduce the Cesáro summable difference sequence space C 1 (Δ) which strictly includes the spaces c 0 (Δ) and c(Δ) but overlaps with (Δ). It is shown that the newly introduced space C 1 (Δ) turns out to be an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of C 1 (Δ) are computed and as an application, the matrix classes ( C 1 (Δ), ), ( C 1 (Δ),c;P) and ( C 1 (Δ), c 0 ) are also characterized.

MSC:40C05, 40A05, 46A45.

1 Notations and definitions

By s we shall denote the linear space of all complex sequences over (the field of complex numbers). , c and c 0 denote the spaces of all bounded, convergent and null sequences x=( x k ) with complex terms, respectively, normed by x = sup k | x k |.

Throughout this paper, unless otherwise specified, we write k for k = 1 and lim n for lim n .

The definitions given below may be conveniently found in [13].

A complete metric linear space is called a Frèchet space. Let X be a linear subspace of s such that X is a Frèchet space with continuous coordinate projections. Then we say that X is an FK space. If the metric of an FK space is given by a complete norm, then we say that X is a BK space.

We say that an FK space X has AK, or has the AK property, if ( e k ), the sequence of unit vectors, is a Schauder basis for X.

A sequence space X is called

  1. (i)

    normal (or solid) if y=( y k )X whenever | y k || x k |, k1, for some x=( x k )X,

  2. (ii)

    monotone if it contains the canonical preimages of all its stepspaces,

  3. (iii)

    sequence algebra if xy=( x k y k )X whenever x=( x k ),y=( y k )X,

  4. (iv)

    convergence free when, if x=( x k ) is in X and if y k =0 whenever x k =0, then y=( y k ) is in X.

The idea of dual sequence spaces was introduced by Köthe and Toeplitz [4] whose main results concerned α-duals; the α-dual of Xs being defined as

X α = { a = ( a k ) s : k | a k x k | <  for all  x = ( x k ) X } .

In the same paper [4], they also introduced another kind of dual, namely, the β-dual (see [5] also, where it is called the g-dual by Chillingworth) defined as

X β = { a = ( a k ) s : k a k x k  converges for all  x = ( x k ) X } .

Obviously, ϕ X α X β , where ϕ is the well-known sequence space of finitely non-zero scalar sequences. Also, if XY, then Y η X η for η=α,β. For any sequence space X, we denote ( X δ ) η by X δ η , where δ,η=α or β. It is clear that X X η η , where η=α or β.

For a sequence space X, if X= X α α then X is called a Köthe space or a perfect sequence space.

A sequence space x=( x k ) of complex numbers is said to be (C,1) summable (or Cesàro summable of order 1) to C if lim k σ k =, where σ k = 1 k i = 1 k x i . By C 1 we shall denote the linear space of all (C,1) summable sequences of complex numbers over , i.e.,

C 1 = { x = ( x k ) s : ( 1 k i = 1 k x i ) c } .

It is easy to see that C 1 is a BK space normed by

x= sup k 1 k | i = 1 k x i | ,x=( x k ) C 1 .

The notion of difference sequence space was introduced by Kizmaz [6] in 1981 as follows:

X(Δ)= { x = ( x k ) s : ( Δ x k ) X }

for X= ,c, c 0 ; where Δ x k = x k x k + 1 for all kN (the set of natural numbers). For a detailed account of difference sequence spaces, one may refer to [718] where many more references can be found.

2 Motivation and introduction

During the last 32 years, a large amount of work has been carried out by many mathematicians regarding various generalizations of difference sequence spaces of Kizmaz. Keeping aside some exceptions (see, for instance, [7, 8]), in most of these works, the underlying spaces have remained the same, i.e., , c and c 0 . In the present work, we take the opportunity to introduce a difference sequence space with underlying space as C 1 .

We observe that

  1. (i)

    C 1 c(Δ) as ( ( 1 ) k ) C 1 but ( ( 1 ) k )c(Δ),

  2. (ii)

    c(Δ) C 1 as (k)c(Δ) but (k) C 1 , and

  3. (iii)

    cc(Δ) C 1 .

Thus the sequence spaces C 1 and c(Δ) overlap but do not contain each other. Similarly, C 1 and also overlap without containing each other as is clear from the fact that C 1 , C 1 and c C 1 . Note that the sequence ( ( 1 ) k 1 k ) is (C,1) summable but not bounded, whereas the sequence x=( x k ) given by x 1 =1, x 2 =0 and

x k ={ 1 , if  2 i 1 < k 3 ( 2 i 2 ) ( i = 2 , 3 , ) ; 0 , otherwise

is bounded but not (C,1) summable. This has motivated the authors to look for a new sequence space which properly includes the spaces C 1 , c(Δ) and .

We now introduce a sequence space C 1 (Δ), Cesàro summable difference sequence space, as follows:

C 1 (Δ)= { x = ( x k ) s : ( Δ x k ) C 1 } .

The overall picture regarding inclusions among the already existing spaces , c, c 0 , C 1 , (Δ), c(Δ), c 0 (Δ) and the newly introduced space C 1 (Δ) is as shown below:

C 1 C 1 ( Δ ) c 0 c c 0 ( Δ ) c ( Δ ) ( Δ ) C 1 ( Δ )

In this paper we show that C 1 (Δ) strictly includes the spaces c 0 (Δ) and c(Δ) but overlaps with (Δ). It is shown that the newly introduced space C 1 (Δ) is an inseparable BK space which does not possess any of the following: AK property, monotonicity, normality and perfectness. The Köthe-Toeplitz duals of C 1 (Δ) are computed, and as an application, the matrix classes ( C 1 (Δ), ), ( C 1 (Δ),c;P) and ( C 1 (Δ), c 0 ) are also characterized.

3 Inclusion theorems and topological properties of C 1 (Δ)

We begin with elementary inclusion theorems justifying that C 1 (Δ) is much wider than , C 1 and c(Δ).

Theorem 3.1 C 1 (Δ), the inclusion being strict.

Proof Let x=( x k ) . Then there exists M>0 such that | x 1 x k + 1 |M for all k1, and so 1 k i = 1 k Δ x i 0 as k. For strict inclusion, observe that (k) C 1 (Δ) but (k) . □

Theorem 3.2 C 1 C 1 (Δ), the inclusion being strict.

Proof For x=( x k ) C 1 , we have lim k 1 k x k =0, and so 1 k i = 1 k Δ x i 0 as k. Inclusion is strict in view of the example cited in Theorem 3.1. □

Theorem 3.3 c(Δ) C 1 (Δ), the inclusion being strict.

Proof Inclusion is obvious since c C 1 . To see that the inclusion is strict, consider the sequence x=( x k )=(1,2,1,2,1,2,). □

Remark 3.4 Let X and Y be sequence spaces. If XY, then X(Δ)Y(Δ).

Proof Since XY, there is a sequence x=( x k )X such that xY. Consider the sequence y=( y k )=(0, x 1 , x 1 x 2 , x 1 x 2 x 3 ,). Then yX(Δ) but yY(Δ). □

Remark 3.5 We have already observed that C 1 and C 1 , so by Remark 3.4, it follows that neither C 1 (Δ) (Δ) nor (Δ) C 1 (Δ). Also, we have c(Δ) C 1 (Δ) (Δ). In view of this and Theorem 3.3, we can say that C 1 (Δ) strictly includes c(Δ) and hence c 0 (Δ) but overlaps with (Δ).

We now study the linear topological structure of C 1 (Δ).

Theorem 3.6 C 1 (Δ) is a BK space normed by

x Δ =| x 1 |+ sup k 1 k | i = 1 k Δ x i | ,x=( x k ) C 1 (Δ).

The proof is a routine verification by using ‘standard’ techniques and hence is omitted.

Theorem 3.7 C 1 (Δ) is not separable.

Proof Let A be the set of all sequences x a , x b , , where

x a = ( k + a ) k =(1+a,2+a,), x b = ( k + b ) k =(1+b,2+b,),

with |ab|> 1 2 ; a,bR. Clearly, A C 1 (Δ) and A is uncountable. Let D be any dense set in C 1 (Δ).

Define a map f:AD as follows:

Let x a A C 1 (Δ). As D is dense in C 1 (Δ), so there exists some z x a D such that x a z x a Δ < 1 4 .

We set f( x a )= z x a .

For x a , x b A, we have

x a x b Δ = | ( 1 + a ) ( 1 + b ) | + sup k 1 k | i = 1 k Δ ( x a x b ) i | | a b | > 1 2 .

Now

z x a x b Δ x a x b Δ x a z x a Δ > 1 2 1 4 = 1 4

and already we have x b z x b Δ < 1 4 , therefore z x a z x b . Hence f is one-to-one. As f(A)D so D is uncountable. Thus, C 1 (Δ) has no countable dense set. □

Corollary 3.8 C 1 (Δ) does not have a Schauder basis.

The result follows from the fact that if a normed space has a Schauder basis, then it is separable.

Corollary 3.9 C 1 (Δ) does not have the AK property.

Theorem 3.10 C 1 (Δ) is not normal (solid) and hence neither perfect nor convergence free.

Proof Taking x=( x k )=(k1) and y=( y k )=( ( 1 ) k (k1)), we see that x C 1 (Δ) but y C 1 (Δ) although | y k || x k |, k1 and so C 1 (Δ) is not normal. It is well known [1] that every perfect space, and also every convergence free space, is normal and consequently C 1 (Δ) is neither perfect nor convergence free. □

Theorem 3.11 C 1 (Δ) is neither monotone nor a sequence algebra.

Proof Take x=( x k )=(k) C 1 (Δ). Consider y=( y k ) where

y k ={ x k , if  k  is odd ; 0 , if  k  is even

i.e., y=(1,0,3,0,5,). Then (Δ y k )=(1,3,3,5,5,) and so (Δ y k ) C 1 , i.e., ( y k ) C 1 (Δ) and hence C 1 (Δ) is not monotone. To see that C 1 (Δ) is not a sequence algebra, take x=y=(k) and observe that x,y C 1 (Δ) but xy=( k 2 ) C 1 (Δ). □

4 Köthe-Toeplitz duals of C 1 (Δ)

In this section we compute the Köthe-Toeplitz duals of C 1 (Δ) and show that C 1 (Δ) is not perfect.

Theorem 4.1

[ C 1 ( Δ ) ] α = { a = ( a k ) : k k | a k | < } = D 1 .

Proof Let a=( a k ) D 1 . For any x=( x k ) C 1 (Δ), we have ( 1 k i = 1 k Δ x i )c, i.e., ( 1 k ( x 1 x k + 1 ))c and so there exists some M>0 such that | x k |M(k1)+ x 1 for k1 and hence sup k k 1 | x k |<, which implies that

k | a k x k |= k ( k | a k | ) ( k 1 | x k | ) <.

Thus, a=( a k ) [ C 1 ( Δ ) ] α .

Conversely, let a=( a k ) [ C 1 ( Δ ) ] α . Then k | a k x k |< for all x=( x k ) C 1 (Δ). Taking x k =k for all k1, we have x=( x k ) C 1 (Δ) whence k k| a k |<. □

Remark 4.2 It is well known [6, 16] that [ c 0 ( Δ ) ] α = [ c ( Δ ) ] α = [ ( Δ ) ] α = D 1 , so we conclude that [ c 0 ( Δ ) ] α = [ c ( Δ ) ] α = [ ( Δ ) ] α = [ C 1 ( Δ ) ] α , i.e., the α-duals of c 0 (Δ), c(Δ), (Δ) and C 1 (Δ) coincide.

Theorem 4.3

[ C 1 ( Δ ) ] α α = { a = ( a k ) : sup k k 1 | a k | < } = D 2 .

Proof Taking m=1 and X=c in [[12], Theorem 2.13], we have [ c ( Δ ) ] α α ={a=( a k ): sup k k 1 | a k |<} and the result follows in view of Remark 4.2. □

Corollary 4.4 C 1 (Δ) is not perfect.

The proof follows at once when we observe that the sequence ( ( 1 ) k (k1)) [ C 1 ( Δ ) ] α α but does not belong to C 1 (Δ).

Theorem 4.5

[ C 1 ( Δ ) ] β = { a = ( a k ) : k k | a k | < } = D 3 .

Proof Let a=( a k ) D 3 and x=( x k ) C 1 (Δ). Then ( 1 k i = 1 k Δ x i )c. For nN, we have

k = 1 n a k x k = k = 2 n (k1) a k ( 1 k 1 i = 1 k 1 Δ x i ) + x 1 k = 1 n a k .

Obviously, ( a k ) and ((k1) a k ) 1 . We define y=( y k ) by y 1 =0 and y k = 1 k 1 i = 1 k 1 Δ x i for all k2. Then yc and since c α = 1 , the series k = 2 (k1) a k ( 1 k 1 i = 1 k 1 Δ x i ) converges absolutely.

Conversely, if a=( a k ) [ C 1 ( Δ ) ] β , then k a k x k converges for all x=( x k ) C 1 (Δ). In particular, taking x k =1 for all k, we have k a k converges and so k = 2 (k1) a k ( 1 k 1 i = 1 k 1 Δ x i ) converges for all x=( x k ) C 1 (Δ). Since x=( x k ) C 1 (Δ) if and only if y=( 1 k i = 1 k Δ x i )c, we have ((k1) a k ) c α . □

Corollary 4.6 [ c 0 ( Δ ) ] α = [ c ( Δ ) ] α = [ ( Δ ) ] α = [ C 1 ( Δ ) ] α = [ C 1 ( Δ ) ] β .

5 Matrix maps

Finally, we characterize certain matrix classes. For any complex infinite matrix A=( a n k ), we shall write A n = ( a n k ) k N for the sequence in the n th row of A. If X, Y are any two sets of sequences, we denote by (X,Y) the class of all those infinite matrices A=( a n k ) such that the series A n (x)= k a n k x k converges for all x=( x k )X (n=1,2,) and the sequence Ax= ( A n x ) n N is in Y for all xX.

The following theorem is well known.

Theorem 5.1 [[3], p.219] Let X and Y be BK spaces and suppose that A=( a n k ) is an infinite matrix such that ( k a n k x k ) n N Y for each xX, i.e., A(X,Y), then A:XY is a bounded linear operator.

Theorem 5.2 A( C 1 (Δ), ) if and only if sup n k = 2 (k1)| a n k |<.

Proof Suppose that sup n k = 2 (k1)| a n k |< and x=( x k ) C 1 (Δ). Proceeding as in Theorem 4.5, we have k = 2 | a n k i = 1 k 1 Δ x i |<.

For mN,

k = 1 m a n k x k = k = 1 m a n k ( i = 1 k 1 Δ x i ) + x 1 k = 1 m a n k ,

which yields the absolute convergence of k a n k x k for each nN, and finally we have

| k a n k x k | ( sup k 2 | 1 k 1 i = 1 k 1 Δ x i | ) ( sup n k = 2 ( k 1 ) | a n k | ) + x 1 sup n k (k1)| a n k |

for all nN.

Conversely, by Theorem 5.1, we have

| k a n k x k | = | A n ( x ) | sup n | A n ( x ) | = A x A x Δ
(5.1)

for each nN and x=( x k ) C 1 (Δ).

Choose any nN and any rN and define

x k ={ ( k 1 ) sgn a n k , if  1 < k r ; 0 , otherwise .

Then x=( x k )c C 1 (Δ) with x Δ =1. Inserting this value of x=( x k ) in (5.1) , we have

k = 2 r (k1)| a n k |A.
(5.2)

Letting r and noting that (5.2) holds for every nN, we are through. □

Remark 5.3 If x=( x k ) C 1 (Δ), then there exists some C such that lim k 1 k i = 1 k Δ x i =. We shall call the C 1 (Δ) limit of the sequence ( x k ) and by ( C 1 (Δ),c;P) we shall denote that subset of ( C 1 (Δ),c) for which C 1 (Δ) limits are preserved.

Theorem 5.4 A( C 1 (Δ),c;P) if and only if

  1. (i)

    sup n k = 2 (k1)| a n k |<,

  2. (ii)

    lim n k (k1) a n k =1,

  3. (iii)

    lim n a n k =0 for each k,

  4. (iv)

    lim n k a n k =0.

Proof Let the conditions (i)-(iv) hold and suppose that x=( x k ) C 1 (Δ) with lim k 1 k i = 1 k Δ x i =. It is implicit in (i) that, for each nN, k (k1)| a n k | converges. It follows that k = 2 (k1) a n k ( 1 k 1 i = 1 k 1 Δ x i ) converges, whence

k a n k x k = k = 2 (k1) a n k ( 1 k 1 i = 1 k 1 Δ x i ) k (k1) a n k + x 1 k a n k .
(5.3)

Let ϵ k = 1 k i = 1 k Δ x i , H= sup k | ϵ k | and M= sup n k (k1)| a n k |.

Then, for any pN, we have

| k = 2 ( k 1 ) a n k ( 1 k 1 i = 1 k 1 Δ x i ) | H k = 2 p (k1)| a n k |+M sup k > p | ϵ k 1 |

and hence

lim sup n | k = 2 ( k 1 ) a n k ( 1 k 1 i = 1 k 1 Δ x i ) | M sup k > p | ϵ k 1 |.

Letting p, we have k = 2 (k1) a n k ( 1 k 1 i = 1 k 1 Δ x i )0 as n. Making use of this and also of (ii) and (iv) in (5.3), we get the result.

Conversely, let A( C 1 (Δ),c;P). Then ( k a n k x k ) n N c for all x=( x k ) C 1 (Δ). By the same argument as in Theorem 5.2, we have sup n k = 2 (k1)| a n k |<. Taking x= e k C 1 (Δ), we get ( a n k ) n N c with lim n a n k =0 for each k. Also, for x=(k1), we have ( k ( k 1 ) a n k ) n N c with lim n k (k1) a n k =1, and finally x=(1,1,1,) C 1 (Δ) yields lim n k a n k =0. □

Theorem 5.5 A( C 1 (Δ), c 0 ) if and only if

  1. (i)

    sup n k = 2 (k1)| a n k |<,

  2. (ii)

    lim n k (k1) a n k =0,

  3. (iii)

    lim n a n k =0 for each k,

  4. (iv)

    lim n k a n k =0.

References

  1. Cooke RG: Infinite Matrices and Sequence Spaces. Macmillan & Co., London; 1950.

    MATH  Google Scholar 

  2. Kamthan PK, Gupta M: Sequence Spaces and Series. Dekker, New York; 1981.

    MATH  Google Scholar 

  3. Maddox IJ: Elements of Functional Analysis. 2nd edition. Cambridge University Press, Cambridge; 1988.

    MATH  Google Scholar 

  4. Köthe G, Toeplitz O: Lineare Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen. J. Reine Angew. Math. 1934, 171: 193–226.

    MathSciNet  MATH  Google Scholar 

  5. Chillingworth HR: Generalized ‘dual’ sequence spaces. Ned. Akad. Wet. Indag. Math. 1958, 20: 307–315.

    MathSciNet  Article  MATH  Google Scholar 

  6. Kizmaz H: On certain sequence spaces. Can. Math. Bull. 1981, 24(2):169–176. 10.4153/CMB-1981-027-5

    MathSciNet  Article  MATH  Google Scholar 

  7. Altay B, Başar F:The fine spectrum and the matrix domain of the difference operator Δ on the sequence space p . Commun. Math. Anal. 2007, 2(2):1–11.

    MathSciNet  MATH  Google Scholar 

  8. Başar F, Altay B: On the space of sequences of p -bounded variation and related matrix mappings. Ukr. Math. J. 2003, 55(1):136–147. 10.1023/A:1025080820961

    Article  MathSciNet  MATH  Google Scholar 

  9. Bektaş ÇA, Et M, Çolak R: Generalized difference sequence spaces and their dual spaces. J. Math. Anal. Appl. 2004, 292: 423–432.

    MathSciNet  Article  MATH  Google Scholar 

  10. Bhardwaj VK, Bala I:Generalized difference sequence space defined by | N ¯ , p k | summability and an Orlicz function in seminormed space. Math. Slovaca 2010, 60(2):257–264. 10.2478/s12175-010-0010-1

    MathSciNet  Article  MATH  Google Scholar 

  11. Çolak R: On some generalized sequence spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 1989, 38: 35–46.

    MathSciNet  MATH  Google Scholar 

  12. Et M, Çolak R: On some generalized difference sequence spaces. Soochow J. Math. 1995, 21(4):377–386.

    MathSciNet  MATH  Google Scholar 

  13. Et M: On some generalized Cesàro difference sequence spaces. Istanb. Üniv. Fen Fak. Mat. Derg. 1996/97, 55/56: 221–229.

    MathSciNet  MATH  Google Scholar 

  14. Et M:Spaces of Cesàro difference sequences of order Δ r -defined by a modulus function in a locally convex space. Taiwan. J. Math. 2006, 10(4):865–879.

    MathSciNet  MATH  Google Scholar 

  15. Et M: Generalized Cesàro difference sequence spaces of non-absolute type involving lacunary sequences. Appl. Math. Comput. 2013, 219(17):9372–9376. 10.1016/j.amc.2013.03.039

    MathSciNet  Article  MATH  Google Scholar 

  16. Malkowsky E, Parashar SD: Matrix transformations in spaces of bounded and convergent difference sequences of order m . Analysis 1997, 17(1):87–97.

    MathSciNet  Article  MATH  Google Scholar 

  17. Malkowsky E, Mursaleen M, Suantai S: The dual spaces of sets of difference sequences of order m and matrix transformations. Acta Math. Sin. 2007, 23(3):521–532. 10.1007/s10114-005-0719-x

    MathSciNet  Article  MATH  Google Scholar 

  18. Orhan C: Cesàro difference sequence spaces and related matrix transformations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 1983, 32(8):55–63.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referee for his/her valuable comments and suggestions, which have improved the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vinod K Bhardwaj.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

VKB and SG contributed equally. All authors read and approved the final manuscript.

An erratum to this article is available at http://dx.doi.org/10.1186/1029-242X-2014-11.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Bhardwaj, V.K., Gupta, S. Cesàro summable difference sequence space. J Inequal Appl 2013, 315 (2013). https://doi.org/10.1186/1029-242X-2013-315

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-315

Keywords

  • sequence space
  • BK space
  • Schauder basis
  • Köthe-Toeplitz duals
  • matrix map